Abstracts Subgroups

This course is an introduction to acylindrically hyperbolic groups through the lens of the special class of hyperbolically embedded subgroups. We will begin with definitions and examples, and then we will investigate some nice geometric (and algebraic) properties enjoyed by these subgroups. We will explore connections with random walks, extending actions, extending quasicocycles, and relative hyperbolicity. Hyperbolically embedded subgroups, particularly those that are virtually cyclic, are a powerful tool for proving important results about acylindrically hyperbolic groups. I will introduce some of these results and sketch the ways in which hyperbolically embeddded subgroups are used in their proof. The course will assume no prior knowledge of any of these topics.

This mini-course will be an introduction to the Bestvina-Brady Morse theory. Let G be a group acting geometrically on a polyhedral complex X. Suppose that φ:G -> Z is an epimorphism, and X admits a φ-equivarient map to a line. The Bestvina-Brady Morse theory relates the finiteness properties of the kernel of φ to topological properties of the link of a vertex in X. We will focus on the applications of the theory in the context of right-angled Artin and Coxeter groups.

After a brief introduction to the general theory of homology of groups we will define l^2 homology via actions on simplicial complexes. In the second lecture we will compute several examples of these homology groups. Finally, we will discuss consequences that are of interest to group theorists particularly in the case that these invariants vanish.

I’ll talk about various ways to think about quasiconvexity in hyperbolic and relatively hyperbolic groups, and explain why and when quasiconvexity persists under Dehn filling.

Given an automorphism of a finitely generate free group, one can construct a corresponding semi-direct product of the free group and the integers. The semi-direct product, also known as a free-by-cyclic group, is finitely generated and we can consider its Cayley graph (or some other geometric model you prefer). We are interested in the dynamics of the defining automorphism, the algebraic properties of the free-by-cyclic group, the coarse geometry of the geometric models, and especially the interplay between these. These ideas are heavily influenced by Thurston’s work on surface homeomorphisms and 3-manifolds. This mini-course will give an incomplete survey of the various results (from old to recent) along these lines and introduce some of the key ideas behind them. There will be no shortage of open problems to take home with you.

We’ll explore the problem of finding ‘nice’ models for the classifying spaces of certain quotients of fundamental groups of non-positively curved cube complexes, and we’ll discuss the framework that provides the necessary tools to do so – cubical small-cancellation theory.

Let GGG be a free product G=A1∗A2∗⋯∗Ak∗FNG = A_1 * A_2* \cdots * A_k * F_NG=A1​∗A2​∗⋯∗Ak​∗FN​, where each of the groups AiA_iAi​ is torsion-free and where FNF_NFN​ is a free group of rank NNN, and let O\mathcal{O}O be the associated deformation space. We show that the diameter of the projection of the subset of O\mathcal{O}O where a given element has bounded length to the Z\mathcal{Z}Z–factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of GGG as a hyperbolic group relative to the collection of subgroups AiA_iAi​ together with a given non-peripheral cyclic subgroup. In a future paper, we will apply this theorem to study the geometry of free group extensions. This is joint work with Caglar Uyanik.

A group G satisfies the Minkowski separation if if has a finite quotient Q in which all finite order outer-automorphisms of G descend as automorphisms of Q and are non trivial in Out(Q). This property was identified as useful in order to compare two relatively hyperbolic groups whose JSJ decomposition is given. Applied to certain subgroups of mapping tori (F \rtimes Z) of a free group F, it allows to compare the conjugacy classes of certain automorphisms of F. In this talk, the groups of interest are the subgroups of suspensions by multi-Dehn twists. (Joint work with N. Touikan.)

The distortion function of a subgroup measures the extent to which the intrinsic word metric of the subgroup differs from the metric induced by the ambient group. Ol’shanskii showed that there are almost no restrictions on which functions arise as distortion functions of subgroups of finitely presented groups. This prompts one to ask what happens if one forces the ambient group to be particularly nice, say, for example, to be hyperbolic. I will survey which functions are known to be distortion functions of subgroups of hyperbolic groups and describe joint work with Tim Riley in which we construct new examples of such functions.

A rational number p/q determines a simple closed curve on a once-punctured torus. When the torus is endowed with a complete hyperbolic metric, each rational gets a well-defined length. If the metric is chosen so that the torus is â€